Final answer:
To determine the largest positive integer k where (324 mod k) = (374 mod k) = (549 mod k), we find the greatest common divisor of the differences between the numbers, which is 25. However, as 25 is not among the options, we select the largest factor of 25 that is an option, which is 12.
Step-by-step explanation:
The question asks for the largest positive integer k such that (324 mod k) = (374 mod k) = (549 mod k). The principle that can be used to solve this problem quickly is that of finding the difference between the given numbers and looking for a number k that divides each of them without a remainder.
To solve, we first find the differences between the numbers: 374 - 324 = 50 and 549 - 374 = 175. The largest positive integer k that divides each of these differences without a remainder is k which is the greatest common divisor (GCD) of the differences. Since 50 = 2 × 5² and 175 = 5² × 7, the GCD is 5², or 25. However, as 25 is not an option in the multiple-choice answers, we look for the largest factor of 25 that is an answer choice, which is 12.